The Chambers The Chambers Separated by Separated by Separated by Separated by Interatrial Interatrial Septum Septum Interventricular Interventricular Septum Septum Right Atrium Right Atrium Blood from Superior and inferior Bl Blood from Superior and inferior venae Bl d f d f S S i i d i f d i f i i venae cavae cavae and the coronary sinus and the coronary sinus d th d th i i Right Ventricle Right Ventricle Receives blood from the right atrium via the right AV valve tricuspid Receives blood from the right atrium via the right AV valve, tricuspid Receives blood from the right atrium via the right AV valve, tricuspid Receives blood from the right atrium via the right AV valve tricuspid valve valve Thin wall Thin wall Left Atrium Left Atrium Receives blood from R and L Pulmonary Veins Receives blood from R and L Pulmonary Veins Left Ventricle Left Ventricle Left Ventricle Left Ventricle Receives blood from the Left AV valve Receives blood from the Left AV valve Thick wall Thick wall Pumps to body via Aortic Pumps to body via Aortic Semilunar Semilunar Valve Valve PASI 2011 - A. Bandoni 10
The Valves The Valves Two types of valves: keep the blood T f l k h bl d flowing in the correct direction. Between atria and ventricles: ca lled atrioventricular valves (also called cuspid valves) p ) Bases of the large vessels leaving the the ventricles: ventricles: called called semilunar semilunar valves . When the ventricles contract When the ventricles contract, atrioventricular valves close to prevent atrioventricular valves close to prevent blood from flowing back into the atria. When the ventricles relax semilunar valves close to prevent blood from When the ventricles relax, semilunar valves close to prevent blood from flowing back into the ventricles. Vales close passively under blood pressure. Responsible for the heart V l l i l d bl d R ibl f th h t sounds. PASI 2011 - A. Bandoni 11
Circulatory System Circulatory System PASI 2011 - A. Bandoni 12
Circulatory System Circulatory System D Deoxygenated blood returns to t d bl d t t the heart via the superior and inferior vena cava, enters the right atrium passes into the right right atrium, passes into the right ventricle, and from here it is ejected to the pulmonary artery. Oxygenated blood returning from the lungs enters the left atrium via the pulmonary veins, passes into the left ventricle, and is then ejected to the aorta. PASI 2011 - A. Bandoni 13
Blood flow pattern through the heart Blood flow pattern through the heart 1. 1.Blood enters right atrium via the superior Blood enters right atrium via the superior and inferior and inferior venae venae cavae cavae 2. 2.Passes tricuspid valve into right ventricle Passes tricuspid valve into right ventricle 3 Leaves by passing pulmonary 3. 3.Leaves by passing pulmonary Leaves by passing pulmonary semilunar Leaves by passing pulmonary semilunar semilunar semilunar valves into pulmonary trunk and to the lungs valves into pulmonary trunk and to the lungs to be oxygenated to be oxygenated 4. 4.Returns from the lung by way of pulmonary Returns from the lung by way of pulmonary veins into the left atrium veins into the left atrium 5. 5.From left atrium past bicuspid valve into left From left atrium past bicuspid valve into left ventricle ventricle 6. 6.Leaves left ventricle past aortic Leaves left ventricle past aortic semilunar semilunar valves into aorta valves into aorta valves into aorta valves into aorta 7. 7.Distributed to rest of the body Distributed to rest of the body PASI 2011 - A. Bandoni 14
Blood flow pattern through the heart Blood flow pattern through the heart PASI 2011 - A. Bandoni 15
Blood Vessels Blood Vessels Blood vessels are divided into a pulmonary circuit and systemic circuit. Blood vessels are divided into a pulmonary circuit and systemic circuit. Artery Artery - - vessel that carries blood away from the heart. Usually vessel that carries blood away from the heart. Usually oxygenated. Exception, pulmonary artery. oxygenated. Exception, pulmonary artery. oxygenated. Exception, pulmonary artery. oxygenated. Exception, pulmonary artery. Vein Vein - - vessel that carries blood towards the heart. Usually vessel that carries blood towards the heart. Usually deoxygenated. Exception pulmonary veins deoxygenated. Exception pulmonary veins Capillary Capillary - - a small blood vessel that allow diffusion of gases, nutrients a small blood vessel that allow diffusion of gases, nutrients and wastes between plasma and interstitial fluid. and wastes between plasma and interstitial fluid. Systemic vessels Transport blood through the body part from left ventricle and back to right atrium g Pulmonary vessels Transport blood from right ventricle through lungs and back to left Transport blood from right ventricle through lungs and back to left atrium Blood vessels and heart are regulated to ensure blood pressure is Blood vessels and heart are regulated to ensure blood pressure is high enough for blood flow to meet metabolic needs of tissues PASI 2011 - A. Bandoni 16
The Real Thing The Real Thing PASI 2011 - A. Bandoni 17
The Real Thing The Real Thing PASI 2011 - A. Bandoni 18
History History History History PASI 2011 - A. Bandoni 19
Mathematical Modelling in Physiology Mathematical Modelling in Physiology With mathematical models it is possible to simulate almost any kind of With mathematical models it is possible to simulate almost any kind of phenomena in nature on a computer. phenomena in nature on a computer. This is a scientific practice of modern science and engineering This is a scientific practice of modern science and engineering (biology, physiology, medicine, biology, physiology, medicine, climate research climate research, ecology, physics, , ecology, physics, chemistry etc ) chemistry etc ) chemistry, etc.) chemistry, etc.) Mathematical modeling in medicine and biology has become so important Mathematical modeling in medicine and biology has become so important that this type of research now has its own name: in that this type of research now has its own name: in silico silico Mathematical modeling undoubtedly will become the paradigm of scientific Mathematical modeling undoubtedly will become the paradigm of scientific Mathematical modeling undoubtedly will become the paradigm of scientific Mathematical modeling undoubtedly will become the paradigm of scientific and medical research in the twenty and medical research in the twenty ‐ first century. first century. In research, the ultimate goal is mechanisms In research the ultimate goal is mechanisms In research the ultimate goal is mechanisms ‐ based models but in reality In research, the ultimate goal is mechanisms ‐ based models, but in reality based models, but in reality based models but in reality models are more often used in a detective models are more often used in a detective ‐ like way to investigate the like way to investigate the consequences of different hypotheses. consequences of different hypotheses. The mathematics modeling is used as a microscope to unveil information The mathematics modeling is used as a microscope to unveil information about reality, that is otherwise inaccessible about reality, that is otherwise inaccessible PASI 2011 - A. Bandoni 20
Heart and Blood Circulation Research History Heart and Blood Circulation Research History Since the dawn of civilization man has been concerned with the Since the dawn of civilization man has been concerned with the Since the dawn of civilization, man has been concerned with the Since the dawn of civilization, man has been concerned with the understanding of living things. understanding of living things. In one of the most ancient medical treatises ( I In one of the most ancient medical treatises (Nei I f th f th t t i i t t di di l t l t ti ti ( (N i N i Ji Nei Jing, 2697 Jing, 2697-2597 BC), Ji 2697 2597 BC) 2697 2597 BC) 2597 BC), blood is mentioned as originating in the heart and distributed in order to blood is mentioned as originating in the heart and distributed in order to return to the starting point. return to the starting point. Despite widespread knowledge of the anatomy of blood vessels, Greeks Despite widespread knowledge of the anatomy of blood vessels, Greeks were unable to find the start of blood circulation by not knowing the were unable to find the start of blood circulation by not knowing the principle of conservation of mass. principle of conservation of mass. The Western world had to wait for William Harvey (1578 The Western world had to wait for William Harvey (1578-1657) to establish The Western world had to wait for William Harvey (1578 The Western world had to wait for William Harvey (1578 1657) to establish 1657) to establish 1657) to establish the concept of circulation. the concept of circulation. PASI 2011 - A. Bandoni 21
History History Di Di Discovery of the closed circulation of blood by William Harvey Discovery of the closed circulation of blood by William Harvey f th f th l l d i d i l ti l ti f bl f bl d b Willi d b Willi H H (1578 (1578 ‐ 1657). 1657). " De Motu Cordis " ("On the Motion of the Heart and Blood“. Frankfurt, 1628) Stroke volume is 70 ml. per beat and Heart beats 72 times per minute, therefore Cardiac Output should be 7.258 liters per day Before 1628, the Before 1628, the Galenic Galenic view of the body prevailed and the concept of view of the body prevailed and the concept of blood circulation was not imaginable. blood circulation was not imaginable. Galen or Galen or Galenius Galenius (Greek physician, II century AD), spent most of his (Greek physician, II century AD), spent most of his lifetime observing the human body and its functioning. lifetime observing the human body and its functioning. Galen believed that the heart acted not as a pump, but rather that it sucked Galen believed that the heart acted not as a pump, but rather that it sucked blood from the veins, that blood flowed from one ventricle to the other of blood from the veins, that blood flowed from one ventricle to the other of the heart through a system of tiny pores of the septum the heart through a system of tiny pores of the septum the heart through a system of tiny pores of the septum. the heart through a system of tiny pores of the septum. Using a simple model, Harvey showed that the amount of blood leaving the Using a simple model, Harvey showed that the amount of blood leaving the heart in a minute could not conceivably be absorbed by the body and h heart in a minute could not conceivably be absorbed by the body and h t i t i i i t t ld ld t t i i bl b bl b b b b d b th b d b th b d b d d d continually replaced by blood made in the liver from continually replaced by blood made in the liver from chyle chyle. . PASI 2011 - A. Bandoni 22
History History Consequently, this model based evidence established the concept that Consequently, this model based evidence established the concept that blood must constantly move in a closed circuit, otherwise the arteries and blood must constantly move in a closed circuit, otherwise the arteries and the body would explode under the pressure. the body would explode under the pressure. the body would explode under the pressure. the body would explode under the pressure. This was discovered about 8 years before the light microscope. This was discovered about 8 years before the light microscope. The concept or method of using mathematical modeling, as a tool for The concept or method of using mathematical modeling, as a tool for making an inaccessible system accessible or an invisible system visible, making an inaccessible system accessible or an invisible system visible, is therefore being coined as “the mathematical microscope” in honor of is therefore being coined as “the mathematical microscope” in honor of f f “ “ f f William Harvey. William Harvey. The mathematical microscope p Ottesen (2011) PASI 2011 - A. Bandoni 23
Th Th The Windkessel The Windkessel Wi dk Wi dk l l Effect Effect PASI 2011 - A. Bandoni 24
The Windkessel Effect The Windkessel Effect The The windkessel windkessel effect is use to describe: effect is use to describe: • Load faced by the heart in pumping blood through pulmonary or systemic arterial system systemic arterial system. • Relation between blood pressure and blood flow in the aorta or pulmonary artery Characteristic parameters of CVS such us compliance and peripheral resistance can be described in terms of the Windkessel models, which is useful in: f l i • Quantifying the effects of vasodilator or vasoconstrictor drugs. • The development and operation of mechanical heart and heart-lung machines. hi Windkessel : a : a german german word that can be translated as air (wind) chamber word that can be translated as air (wind) chamber Windkessel (k (kessel k kessel). l l) ). First description by German physiologist Otto Frank in 1899. p y p y g PASI 2011 - A. Bandoni 25
The Windkessel Effect The Windkessel Effect Heart and systemic arterial system similar to a closed hydraulic circuit Heart and systemic arterial system similar to a closed hydraulic circuit comprised of a water pump connected to a chamber. comprised of a water pump connected to a chamber. The circuit is filled with water except for a pocket of air in the chamber The circuit is filled with water except for a pocket of air in the chamber Arterial compliance Peripheral P i h l ressistance As water is pumped into the chamber, the water both compresses the air in As water is pumped into the chamber, the water both compresses the air in the pocket and pushes water out of the chamber the pocket and pushes water out of the chamber the pocket and pushes water out of the chamber. . the pocket and pushes water out of the chamber. . PASI 2011 - A. Bandoni 26
The Windkessel Effect The Windkessel Effect The compressibility of the air in the pocket simulates the The compressibility of the air in the pocket simulates the elasticit elasticity and y and extensibility extensibility of the major artery, as blood is pumped into it by the heart of the major artery, as blood is pumped into it by the heart ventricle. ventricle. This effect is commonly referred to as arterial compliance This effect is commonly referred to as arterial compliance. . The The resistance resistance water encounters while leaving the Windkessel, simulates water encounters while leaving the Windkessel, simulates the resistance to flow encountered by the blood as it flows through the the resistance to flow encountered by the blood as it flows through the arterial tree from the major arteries to minor arteries to arterioles and to arterial tree from the major arteries to minor arteries to arterioles and to arterial tree from the major arteries, to minor arteries, to arterioles, and to arterial tree from the major arteries, to minor arteries, to arterioles, and to capillaries, due to decreasing vessel diameter. capillaries, due to decreasing vessel diameter. This resistance to flow is commonly referred to as peripheral resistance This resistance to flow is commonly referred to as Thi Thi i t i t t t fl fl i i l l f f d t d t peripheral resistance. i h i h l l i t i t PASI 2011 - A. Bandoni 27
The Windkessel Effect The Windkessel Effect Hypotheses : Hypotheses : • Unsteady flow. • The pressure diff. across the resistance is a linear function of the flow rate t • The working fluid is incompressible (constant air pressure to volume ratio) • The flow is constant throughout the ejection phase. The Windkessel 2-elements considers only the arterial compliance (C) and y p ( ) the peripheral resistance (R). Symbols: Symbols: P : pressure generated by the heart (N.m -2 ) [mmHg] Q : blood flow in the aorta (m 3 .s -1 ) [l.mn -1 ] R : peripheral resistance (N s m -5 ) [dyne s cm -5 ] R : peripheral resistance (N.s.m 5 ) [dyne. s.cm 5 ] C : arterial or systemic compliance (m 5 .N -1 ) [ml.mmHg -1 ] t : time [(s) T T : period (s) i d ( ) Ts: ejection time (s) PASI 2011 - A. Bandoni 28
The Windkessel Effect The Windkessel Effect Theoretical development of the Windkessel effect air Q T s V(t) P(t) R Q (t) Q 1 (t) Q(t) Q(t) P cv P t T Schematic representation Schematic representation Systolic phase: of a chamber Diastole phase: valve in open valve in close position position position position PASI 2011 - A. Bandoni 29
The Windkessel Effect The Windkessel Effect I - Systolic phase (valve in open position) ( l i iti ) I S t li h t 0 T s Conservation of mass: Q cc : flow to the compliance chamber Q Q Q in out cc dV Thus: P cv : central venus pressure: ( P cv << P ) Q Q 1 dt ( P cv ≅ 5 mmHg vs. P ≅ 100 mmHg ]) Hyp.4: Q = Cte. throughout the systolic phase, thus: cv P P R . Q 1 P dV P dV dP Therefore: Compliance (C) Q . R dt R dP dt dP ( t ) P ( t ) Q ( t ) P ( t ) dP ( t ) Then: or Q ( t ) C . dt dt R R . C C C C R R dt dt PASI 2011 - A. Bandoni 30
The Windkessel Effect The Windkessel Effect Solution of the differential equation t a) Particular solution (Q = Cte.=0) P ( t ) . exp( ) 1 R . C b) Method of variation of parameter ( α 1 = α 1 (t) ) d t 1 t Q ( t ). exp( ) ( t ). exp( ) 1 1 dt R . C R . C R . C C 1 t t d ( t ) 1 t Q 1 . ( t ). exp( ) exp( ) ( t ). exp( ) 1 1 R . C R . C R . C dt R . C R . C C d ( t ) Q t t Hence: Then: Then: 1 . . e p( exp( ) ) ( ( t t ) ) R R . . Q Q . . exp( exp( ) ) 1 1 2 2 dt C R . C R . C PASI 2011 - A. Bandoni 31
The Windkessel Effect The Windkessel Effect c) The general solution for systolic phase is t t t t P ( t ) R . Q . exp( ) . exp( ) s 2 R . C R . C To determine α 2 we can use initial condition P (t=0)= P 0 , then α 2 = P 0 - R.Q P ( t 0 ) P P R . Q 0 2 0 Finally, the pressure waveform for the systolic phase can be written as t P ( t ) R . Q P R . Q . exp( ) s 0 R . C PASI 2011 - A. Bandoni 32
The Windkessel Effect The Windkessel Effect I – Diastolic phase (valve in close position) ( l i l iti ) I Di t li h T t T s air Following similar reasoning but with Q =Cte.=0 V(t) P(t) dP dP P P Q Q Q 1 (t) dt R . C C With initial condition: P (t= T s )= P s ( T s ), the solution to the differential equation is: t t where ( ( ) ) . exp( p( ) ) P t P exp( p( ) ) 1 . R . Q Q . 3 3 3 3 0 0 R R . C C R . C Fi Finally, the pressure waveform for the diastolic phase can be written as: ll th f f th di t li h b itt t t P ( ( t ) ) P exp( p( ) ) 1 . R . Q Q . exp( p( ) ) d d 0 0 R . C R . C PASI 2011 - A. Bandoni 33
The Windkessel Effect The Windkessel Effect C Complete model l t d l t Systolic Phase 0 T s air V(t) t P(t) P P ( ( t t ) ) R R . Q Q P P R R . Q Q . exp( exp( ) ) s 0 . R C Q 1 (t) T t T Diastolic Phase air s V(t) P(t) t t t t P ( t ) P exp( ) 1 . R . Q . exp( ) d 0 R . C R . C Q 1 (t) Given: T s exp( ) 1 R . C . . . . P P R R Q Q Q Q , R R , C C , T T , T T and and P P ( ( data data ) ) or or 0 0 s 0 T 1 exp( ) R . C PASI 2011 - A. Bandoni 34
The Windkessel Effect The Windkessel Effect Th The term R . C it is crucial in the 2-W because it determine the “speed” t R C it i i l i th 2 W b it d t i th “ d” of the exponential decay. This product is called the “characteristic time”, called P P P P R.Q P 0 0 t t Case: 0 0 Case: Case: Case: PASI 2011 - A. Bandoni 35
The Windkessel Effect The Windkessel Effect Hypertension: P s > 140 mmHg 0 , Case: P d > 90 mmHg PASI 2011 - A. Bandoni 36
The Windkessel Effect The Windkessel Effect The electrical circuit equivalence Basic equation of a 2-element Winkessel model: q P ( t ) dP ( t ) Q Q ( ( t t ) ) C C . R dt Electric circuit of 2 passive elements: Electric circuit of 2 passive elements: I(t) I(t) : electrical current l t i l t E(t) : electrical potential I(t) I 3 C : capacitance of the capacitor R : resistance of the resistor I 2 From the Ohm and Kirchhoff laws E ( t ) dE ( t ) R E(t) C I ( t ) C . R dt I(t) ≡ Q(t) (blood flow) E(t) ≡ P(t) (arterial blood pressure) C C ≡ C (arterial compliance) ≡ C (arterial compliance) R ≡ R (peripheral resistance) PASI 2011 - A. Bandoni 37
The Windkessel Effect The Windkessel Effect The 3-element Windkessel model R 2 I(t) I( ) I(t) ≡ Q(t) (blood flow) Q(t) (bl d fl ) E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance) R 1 ≡ R 1 (peripheral resistance R C (syst. and pulm.circuits)) E(t) 1 R 2 ≡ R 2 (resistance of valves 2 ( 2 (aortic and pulmonary)) R dE ( ( t ) ) P ( ( t ) ) dP ( ( t ) ) 1 1 1 1 . I I ( ( t t ) ) C C . R R . C C . 1 R dt R dt 2 2 PASI 2011 - A. Bandoni 38
The Windkessel Effect The Windkessel Effect The 4-element Windkessel model R 2 I(t) I(t) ≡ Q(t) (blood flow) I(t) Q(t) (blood flow) E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance) R 1 R ≡ R (peripheral resistance ≡ R 1 (peripheral resistance (syst. and pulm.circuits)) R 1 C E(t) E(t) R 2 ≡ R 2 (resistance of valves (aortic and p lmonar )) (aortic and pulmonary)) L ≡ L (inertia of the blood circulation) L 2 R L dE ( t ) d E ( t ) P ( t ) dP ( t ) 1 1 1 . I I ( ( t t ) ) R R . C C . L L . C C . C C . 1 2 R R dt dt R dt 2 2 2 PASI 2011 - A. Bandoni 39
Compartment Compartment Compartment Compartment M d l M d l Models Models PASI 2011 - A. Bandoni 40
Compartment Models Compartment Models They are used to describe transport material in biological sciences They are used to describe transport material in biological sciences They are used to describe transport material in biological sciences They are used to describe transport material in biological sciences A compartment model contains a certain number of compartments, each A compartment model contains a certain number of compartments, each one with a well mixed material one with a well mixed material one with a well mixed material one with a well mixed material Compartments exchange material following certain rules Compartments exchange material following certain rules Material can be stored in the boxes and transported between them Material can be stored in the boxes and transported between them Every compartment has a number of connections entering and leaving it. Every compartment has a number of connections entering and leaving it. Material can be added from the outside, can be removed or transported. Material can be added from the outside, can be removed or transported. Source Drain PASI 2011 - A. Bandoni 41
Compartment Models Compartment Models Material represent the amount of something that we wish to account for Material represent the amount of something that we wish to account for Material represent the amount of something that we wish to account for Material represent the amount of something that we wish to account for To account for the material, the models must fulfill certain conservation To account for the material, the models must fulfill certain conservation laws. laws. Conservations laws state that the difference between input and output Conservations laws state that the difference between input and output Conservations laws state that the difference between input and output Conservations laws state that the difference between input and output flows amounts how much will be stored. flows amounts how much will be stored. A compartment model can also represent: A compartment model can also represent: A compartment model can also represent: A compartment model can also represent: Ecological systems (material could be energy and the compartment Ecological systems (material could be energy and the compartment different species of animals or plants) different species of animals or plants) Physiologic system (material could be oxygen and compartment de Physiologic system (material could be oxygen and compartment de organs) organs) Compartment can not be thought as independent. Flow in and out may Compartment can not be thought as independent. Flow in and out may depend on the compartment volume depend on the compartment volume Inflow to compartment may depend of outflow of other compartment. Inflow to compartment may depend of outflow of other compartment. PASI 2011 - A. Bandoni 42
Compartment Models Compartment Models State variables depend on each other and on the state of the system as a State variables depend on each other and on the state of the system as a State variables depend on each other and on the state of the system as a State variables depend on each other and on the state of the system as a whole. whole. The transport in and out is characterized by the flows velocities. The transport in and out is characterized by the flows velocities. Limitations of the compartment model Limitations of the compartment model Limitations of the compartment model Limitations of the compartment model Is the system closed . Equation of conservation of mass is correct . Equation of conservation of mass is correct Is the system closed • only if all material added or removed is included in the model. There only if all material added or removed is included in the model. There is some lost of detailed information is some lost of detailed information is some lost of detailed information. is some lost of detailed information. Homogeneity assumption . Not always it is possible to keep this . Not always it is possible to keep this Homogeneity assumption • assumption. Then more compartments are needed but also more assumption. Then more compartments are needed but also more information it is required information it is required information it is required. information it is required. Accuracy of the balance equation . In real physiological system . In real physiological system Accuracy of the balance equation • typically some mass balance are know and other are not. typically some mass balance are know and other are not. Relevance of the mass balance . Not all systems can be described . Not all systems can be described N t N t ll ll t t b b d d ib d ib d Relevance of the mass balance R l R l f th f th b l b l • in terms of mass balances. in terms of mass balances. Sensitivity analysis . Initial conditions and . Initial conditions and model parameters are model parameters are not not Sensitivity analysis • always known precisely. always known precisely. PASI 2011 - A. Bandoni 43
Mathematical Mathematical Mathematical Mathematical M d l M d l Models Models Cardiovascular, Respiratory Cardiovascular, Respiratory and Pharmacodynamic and Pharmacodynamic and Pharmacodynamic and Pharmacodynamic PASI 2011 - A. Bandoni 44
Human Circulatory System Model Human Circulatory System Model The The historical historical fascination fascination of of the the heart heart has has lasted lasted for for many many centuries centuries and and continues to continues to attract attract considerable considerable attention attention both both theoretically theoretically and and clinically clinically. . To To develop develop a a physiologically physiologically founded founded model model of of the the heart heart and and the the vasculature, vasculature, it it is is essential essential to to have have a a good good model model of of the the human human short short term term press re pressure control pressure press re control control represented control represented represented b the represented by by the the baroreceptor the baroreceptor baroreceptor mechanism baroreceptor mechanism mechanism mechanism. Using Using Using Using a a lumped lumped lumped lumped parameter parameter parameter parameter compartment compartment compartment compartment model, model, model, model, the the the the entire entire entire entire human human human human cardiovascular cardiovascular system system may may be be described described as as a a network network of of compliances, compliances, resistances resistances and and inductances inductances not not reflecting reflecting anatomical anatomical properties properties. . Although Although strikingly strikingly simple, simple, the the model model gives gives a a very very good good description description of of the the input input impedance impedance of of the the arterial arterial system system. . Such Such models models are are valuable valuable tools tools for for understanding understanding cardiovascular cardiovascular diseases diseases (hypertension (hypertension weak (hypertension, weak (hypertension, weak and weak and and enlarged and enlarged enlarged heart enlarged heart, heart, hemorrhages, heart hemorrhages hemorrhages, etc hemorrhages etc ) etc etc.) PASI 2011 - A. Bandoni 45
Human Circulatory System Model Human Circulatory System Model Models Models facilitates facilitates getting getting new new insight insight into into cardiovascular cardiovascular functions functions and and the the interaction interaction with with other other system system (central (central nervous nervous system, system, respiratory respiratory systems, systems, etc etc. .) This This type type of of models models can can be be reliable reliable and and stable, stable, simply simply enough enough to to run run in in real real ti ti time time. . Lumped cardiovascular Lumped Lumped Lumped cardiovascular cardiovascular models cardiovascular models models are models are are divided are divided divided into divided into into pulsatile into pulsatile pulsatile and pulsatile and and non and non non pulsatile non-pulsatile pulsatile. pulsatile In In the the pulsatile pulsatile case, case, the the heart heart functioning functioning is is guided guided by by a a time time- -varying varying elastance elastance function function. . A lumped A lumped lumped pulsatile lumped pulsatile pulsatile cardiovascular pulsatile cardiovascular cardiovascular model cardiovascular model model that model that that embraces that embraces embraces principal embraces principal principal features principal features features features of of the the human human circulation circulation. . PASI 2011 - A. Bandoni 46
Human Circulatory System Model Human Circulatory System Model Lumped Lumped cardiovascular cardiovascular models models are are divided divided into into pulsatile pulsatile and and non non- -pulsatile pulsatile. . In the I In th the pulsatile th pulsatile case, l l til til case, the th th the heart heart functioning h h t t f f functioning is ti ti i i i i is guided guided by id d b id d by a a time ti ti time- -varying varying i i elastance elastance function function. . A lumped lumped pulsatile pulsatile cardiovascular cardiovascular model model that that embraces embraces principal principal features features of of the the human human circulation circulation. . PASI 2011 - A. Bandoni 47
Human Circulatory System Model Human Circulatory System Model Pulmonar circulation Ap3 Ap2 Vp1 Ap1 Vp2 LA LA RV RV Heart LV RA As1 Vs2 As2 Vs1 Systemic As3 circulation PASI 2011 - A. Bandoni 48
Human Circulatory System Model Human Circulatory System Model Pp3 Cp3 Vp3 Rp3 Ap3 Qp2 Qp3 Cp2 Pp2 Pl1 Cl2 Ap2 Vp1 Rp2 Vp2 Vl1 Rl2 Qp1 Ql1 Cl2 Cp1 Cp1 Pl2 Pl2 Pp1 P 1 Ap1 Rl2 Vp2 Rp1 Vl2 Vp1 Ll2 Lp1 Qrv PV Ql2 Ela Pla Eminrv Rla Vla LA Erv(t) Prv RV RV Emaxrv Emaxrv Lla Lla Lrv Vrv MV Qla TV Qra Era Eminlv LV Pra Plv Elv(t) RA Rra Emaxlv Vra Vlv Llv Lra AV AV Qv2 Qv2 Qlv Qlv Cv2 Ca1 Pv2 As1 Pa1 Vs2 Rv2 Ra1 Vv2 Va1 Lv2 La1 Qa1 Qv1 Cv1 Pv1 Pa2 Ca2 As2 Vs1 Rv1 Vv1 Va2 Ra2 As3 Qa3 Qa2 Pa3 Ca3 Va3 Ra3 PASI 2011 - A. Bandoni 49
Human Circulatory System Model Human Circulatory System Model Model of a typical compartment (chamber) of the hemodynamic system V 0 : volumen at p=0 R : ressistance L : inertia Hemodynamic y C C : compliance li element of a p i p 0 blood chamber Blood Blood input input output output Q in Q out R L p i p 0 Equivalence with an electric Q Q out C V 0 circuit Q in PASI 2011 - A. Bandoni 50
Circulatory System Model (Ottesen et al., 2003) • Heart Model o Heart itself: 4 chambers (2 atria and 2 ventricles) H t it lf 4 h b (2 t i d 2 t i l ) o Vascular part Systemic part: 5 chambers (systemic arteries and veins) Pulmonary part: 5 chambers (arteries and veins) Pulmonary part: 5 chambers (arteries and veins) • Baroreceptor Model o Chronotropic effect (on heart rate) o Inotropic effect (on the cardiac contractility) o Vascular effect (on arteries and veins) V l ff t ( t i d i ) Respiratory System Model (Christiansen and Dræby, 1996) • Lung Model Lung Model o Upper respiratory tracks: 1 chamber o Alveoli: 1 chamber • Gas Transport in Blood Model (O2, CO2, Anesthesia) o Vascular part: 5 chambers V l t 5 h b o Organs and tissues: 8 compartments Organs compartments: one part of tissue and one part of blood (equilibrium of the substances distributed by the blood on both sides it is assumed) of the substances distributed by the blood on both sides it is assumed) It is assumed constant blood (VB) and tissue (VT) volumes. o Capillaries and alveoli: 1 chamber Ph Pharmacodynamic Model (Gopinath et al., 1995) d i M d l • Drug Effect on Hemodynamic Variables Model PASI 2011 - A. Bandoni 51
The Cardiovascular Model The Cardiovascular Model The Pumping Heart The Pumping Heart Based Based on on an an elastance model where where the the cardiac cardiac elastance model contraction contraction properties properties of of the the two two ventricles ventricles are are representing representing by representing representing by by a pair by a pair pair of pair of of time of time time varying time-varying varying elastance varying elastance elastance elastance functions functions. . The inertia The The inertia The blood movements blood movements movements in movements in in the in the the ventricles the ventricles ventricles is ventricles is is considered is considered considered through considered through through through inertia of inertia of of blood of blood an an inductance inductance that that introduce introduce a a phase phase shift shift between between the the ventricular ventricular pressure pressure and and the the root root aortic aortic pressure pressure. . The The viscous blood in in the the two two atria atria are are included included by by viscous properties properties of of blood ventricular ventricular filling filling resistance g resistance PASI 2011 - A. Bandoni 52
Q l Q l p lv L la M AV R la v a L lv p as p la V LA LA LV LV Left Left E la E E lv (t) Heart R os R a1 L a1 R a2 R a3 R v1 R v2 L v2 p a3 p a1 p a2 p v1 p v2 p as Syst. AA AA CVi CVi Circ. C a2 C a3 C v1 C v2 C a1 Q r Q rv L ra TV R rv PV R ra a p rv p ap p ra Right Heart RV RA E ra E rv (t) R p1 R a3 R op L p1 R p2 R l1 R l2 L l2 PV p l2 p p2 p p3 p l1 p p1 Pulm . PA PA CVs CVs C p1 C p2 C p3 C l1 C l2 Circ. PASI 2011 - A. Bandoni 53
The Pumping Heart The Pumping Heart dQ p p R . Q if p p la la lv la la la lv dt L la if if p p Q 0 la lv la dV la Q Q Q Q l 2 la dt p E . V V la la la d , la t V V Q Q dt dt 2 2 ml ml lv , b lv * t dQ p p E t E . 1 t E . t if p p lv lv as lv min, lv max, lv lv as d dt L L . t 2 . . t lv if p p a . sin b . sin , 0 t t Q 0 lv as ce lv t t t ce ce dV dV 0 , t t t lv Q Q ce h la lv dt t . t 0 1 ce h p p E E ( ( t t ). ) V V V V lv lv lv d , lv p R . Q p as 0 s lv as PASI 2011 - A. Bandoni 54
The Pumping Heart The Pumping Heart Elastance model E max,lv E min,lv t ce ce t h . t 2 . . t a . sin b . sin , 0 t t E t E . 1 t E . t ce t t t lv min, lv max, lv ce ce 0 , t t t ce ce h h PASI 2011 - A. Bandoni 55
The Circulatory System Model The Circulatory System Model Single chamber model pa1 Qa1 Q 1 dV dV a 2 pa2 Q Q a 1 a 2 dt Va2 V V V V a 2 un , a 2 p a 2 C a 2 Qa2 p p p p a 2 a 3 Q a 2 R a 2 PASI 2011 - A. Bandoni 56
The Baroreceptors Model The Baroreceptors Model Baroreceptors (BR) are sensors of mean blood pressure that are located in the blood vessels of several mammals. BR nerves are stretch receptors which responds to changes in blood pressure. BR can send messages to the CNS to increase or decrease total peripheral resistance and cardiac output (CO). BR act immediately as part of a negative feedback system called the baroreflex, returning mean arterial blood pressure (MAP) to a normal level as soon as there is a change as soon as there is a change. BR detect the amount of stretch of the blood vessel walls, and send the signal to the CNS system in response to this stretch. A hysteresis-like phenomena is observed: the response to a pressure increase is different to the response to a pressure-decrease PASI 2011 - A. Bandoni 57
The Baroreceptors Model The Baroreceptors Model ① Increased blood pressure stretched carotid arteries and aorta causing the baroreceptor to increase their basal rate of action potential generation. ② A ti ② Action potential are conducted by t ti l d t d b the glossopharyngeal and the vagus nerves to the cardioregulatory and vasomotor centers in the medulla t t i th d ll oblongata. ③ As a result of increased ③ As a result of increased stimulation from the baroreceptor, the cardioregulatory center increased parasymphatic stimulation to the parasymphatic stimulation to the heart, which decreases the heart rate. ④ Also, as a result of increased stimulation from the baroreceptor, the ④ Also, as a result of increased stimulation from the baroreceptor, the cardiorvascular center decreases sympathetic stimulation to the heart, which decreases heart rate stroke volume. PASI 2011 - A. Bandoni 58
The Baroreceptors Model The Baroreceptors Model ⑤ The vasomotor center decreases sympathetic stimulation to blood vessels, causing vasodilatation. The vasodilatation along with the decreased heart rate and decreased stroke volume bring the elevated blood pressure back toward normal. If the initial problem were decrease in blood pressure, f the activities and effect of baroreceptors, cardiovascular center and vasomotor center would be opposite of what was illustrated. it f h t ill t t d PASI 2011 - A. Bandoni 59
The Baroreceptors Model The Baroreceptors Model Baroreceptor H Heart system system frequency E maxlv , E maxrv Systolic maximum maximum elastance R a1 , R a2 , R a3 Cardio- MAP Systemic vascular vascular resistance resistance arteries System Compliance C v1 , C v2 in veins and in veins and arteries Unstressed V unv1 , V unv2 vol in syst vol. in syst. veins PASI 2011 - A. Bandoni 60
The Baroreceptors Model The Baroreceptors Model Afferent sector Efferent sector n s MAP Central x i n Eferent n p Nervous Sensors Sensors pathways th System 1 b n s MAP MAP . n . MAP . n . MAP i i s i p i MAP MAP 1 dx t 1 b i , x t MAP i E i i dt dt 1 i n p MAP MAP 1 i i E E H H , E E , R R , V V , C C max ps un v PASI 2011 - A. Bandoni 61
The Respiratory System Model The Respiratory System Model The The respiratory respiratory system p p y y system is y is concerned concerned with with the the transport transport of p of oxygen oxygen yg yg between between atmosphere atmosphere and and the the tissue tissue and and organs organs in in the the body body Oxygen Oxygen is O i i is continuously continuously transported ti ti l l t t transported by t d b th t d by the the lung th lung and l l and blood d bl blood circuit bl d i d circuit. i it it Carbon Carbon Carbon dioxide Carbon dioxide dioxide is dioxide is is a waste is a waste waste product waste product product of product of of the of the the oxidative the oxidative oxidative metabolism oxidative metabolism metabolism and metabolism and and is and is is is carried by carried by the the blood blood in in the the opposite opposite direction direction PASI 2011 - A. Bandoni 62
The Respiratory System Model The Respiratory System Model O 2 CO 2 Atmosphere Ventilation Ventilation O 2 CO 2 Alveoli Gas exchange O 2 CO 2 Pulmonary circulation Ri ht Right L ft Left Gas transport Heart Heart Systemic circulation O 2 CO 2 Gas exchange Cell metabolism PASI 2011 - A. Bandoni 63
The Respiratory System Model The Respiratory System Model Lung model: pressure Lung model: pressure ■ Connect atmosphere (mask) ■ Connect atmosphere (mask) with alveoli trought expressions of gas flow R 0 Alveoli ■ The lung is divided in compartments Upper R 1 R 1 R 2 R 2 R i R i airway i ■ In each compartment gas C 0 C 1 C 2 C i flows are calculated (O2, U m (t) CO2, Anesthesia) Atmosphere or ■ The outputs of the model are: U t (t) respiratory i t pressure in different sectors, Muscles mask the net volume of air flow, partial pressure of expired air pa t a p essu e o e p ed a and alveoli. PASI 2011 - A. Bandoni 64
The Respiratory System Model The Respiratory System Model Distribution of substances in the organs through blood Distribution of substances in the organs through blood Alveolus Capillary Alveolus κ .p A κ .p p cp Q Q.c b (1- λ ).Q.c vs ( ) Q V b c b (p) V (p) b vs C Central t l Central p v arterial Metabolism p as venous λ .Q.c vs s compartment compartment p li Liver Kidney p ki Metabolism Viscere M + M - c vv p he Heart venous venous compartment Brain p br V t c t (p) Other p re organs V b c b (p) V b c b (p) z i .Q.c as p co Lean venous Connective z i .Q(p).c b c vl compartment tissue p mu Muscles Adi Adipose tissue venous c av p ad Adipose tissue compartment PASI 2011 - A. Bandoni 65
The Respiratory System Model The Respiratory System Model Upper dpi n Alveoli U p R . C . airways p 0 dp m 0 0 i i 1 dt dt 0 R 0 R R 0 . C 0 C 0 p i dp p p U f 0 R i C i i 0 i t , , i 1 ... n p p cp A A d dt R i C . i f i Pressure model f R f f f f . . d T I U p i I p p n 0 m 0 e 0 i 0 i 0 2 i 1 1 dt dt C C p p V V p p R R R R 0 0 00 0 0 i R f f f d T I p U p . κ p f i 0 t i 0 i . p cp cp i i i i 2 2 . . dt dt C C p V V p R R i i 0 i i i 0 x 0 I x x x x 0 Molar fractions model 66 PASI 2011 - A. Bandoni
The Pharmacodynamic Model The Pharmacodynamic Model Pharmacology : the the history, history, source, source, physical physical and and chemical chemical properties, properties, Pharmacology biochemical biochemical and and physiological physiological effect, effect, mechanisms mechanisms of of action, action, absorption, absorption, distribution, distribution, biotransformation biotransformation and and excretion, excretion, and and therapeutic therapeutic and and other other uses uses of of drugs drugs. . Pharmacokinetics : absorption, Pharmacokinetics : absorption absorption, distribution, absorption distribution distribution, metabolism distribution metabolism metabolism and metabolism and and excretion and excretion excretion of excretion of of of Pharmacokinetics Pharmacokinetics drugs drugs. . Pharmacodynamics : biochemical biochemical and and physiological physiological effects effects and and their their Pharmacodynamics mechanisms mechanisms of of action action. . PASI 2011 - A. Bandoni 67
The Pharmacodynamic Model The Pharmacodynamic Model tion ncentrat Drug Co D Time Concentration of drug in the body as a function of time Concentration of drug in the body as a function of time PASI 2011 - A. Bandoni 68
The Pharmacokinetic Model The Pharmacokinetic Model 1 p c c d d d c c c c p p M M p p M M p p t b V V V V z z Q Q as b b t t b b i i p p dt d d Pressure balance 0 M i in the organs th M c CO 2 M O M 2 M 0 O 2 c O O 2 2 0 c aa M M aa c aa aa 1 p c Pressure balance in d d c c p p p b V Q 1 s b A b p v the capillaries dt d 1 Pressure balance in c p c c d d Q Q c c p b c 1 1 2 2 V Q the compartments p b p Q b p x x dt dt d d Q Q Q 1 2 PASI 2011 - A. Bandoni 69
The Pharmacodynamic Model The Pharmacodynamic Model C C d Drug effect E maxBARO E max = E maxBARO (1 ± Eff) MAP R sisBARO R= R sisBARO (1 ± Eff) Baro- receptors receptors dEff N k . C . Eff Eff k . Eff 1 d max 2 dt dEff dC Cardiovascular C a 1 C . a 1 PFL system a 1 BASE dt dt dEff dE dE E max lv max lvBARO 1 1 Eff Eff E E max lvDP E max E max l BARO lvBARO dt dt lvDP dt dEff dEff dR dR R R sis sisBARO 1 Eff ff Eff ff R sisDP sisSNP R R R R sisBARO sisBARO dt dt dt dt sisDP sisSNP PASI 2011 - A. Bandoni 70
The Pharmacodynamic Model The Pharmacodynamic Model Drug (intravenous) Affected variable Action SNP (sodium SNP (sodium Peripheral resistance MAP nitroprusside) Peripheral resistance, DP (dopamine) ( p ) MAP systolic maximum elastance systolic maximum elastance MAP PFL (propofol) BIS unconsciousness Systolic maximum elastance Peripheral resistance PASI 2011 - A. Bandoni 71
The Pharmacodynamic Model The Pharmacodynamic Model DP DP and and SNP SNP drugs drugs are are chosen chosen to to increase increase ventricular ventricular contractility contractility and and reduce reduce the d th the resistance th resistance of i t i t of arteries f arteries to t t i i to blood t blood flow, bl bl d fl d flow, respectively fl respectively. ti ti l l . PFL PFL is PFL PFL is is chosen is chosen chosen to chosen to to conduct to conduct conduct unconsciousness conduct unconsciousness unconsciousness by unconsciousness by by measured by measured measured of measured of of BIS of BIS BIS BIS parameter parameter. . DP DP increases increases the the MAP MAP and and CO CO. . SNP SNP decreases decreases and and increases increases CO CO MAP MAP. . Sceneries are simulated by delivering a step of 1 μ g/kg/min of SNP, DP Sceneries are simulated by delivering a step of 1 μ g/kg/min of SNP, DP and PFL and registering the dynamic response of the physiological, and PFL and registering the dynamic response of the physiological, pharmacokinetic and pharmacod namic ariables pharmacokinetic and pharmacod namic ariables pharmacokinetic and pharmacodynamic variables. pharmacokinetic and pharmacodynamic variables. PASI 2011 - A. Bandoni 72
Computational Implementation Computational Implementation Model implemented in Fortran Model implemented in Fortran Diff. Diff. Eqs Eqs. solved with a 4th order . solved with a 4th order Runge Runge- -Kutta Kutta method. method. Resolution sequence: Resolution sequence: (i i) ) the cardiovascular model is solved until to the cardiovascular model is solved until to reach steady state, reach steady state, (ii) (ii) the CO obtained from this model is used in the the CO obtained from this model is used in the breathing model, breathing model, (iii) g g , ( ) , ( ) (iii) the breathing model is solved until to reach steady the breathing model is solved until to reach steady g g y y state. state. The drug injection is simulated The drug injection is simulated for a cycle of breathing (5 sec.) Th Th d d i j i j ti ti i i i i l t d l t d f for a cycle of breathing (5 sec.). Then the f l l f b f b thi thi (5 (5 ) ) Th . Then the Th th th cardiovascular model is fed with the drug concentration cardiovascular model is fed with the drug concentration Cd Cd to simulate to simulate the the 0.8 sec. a heartbeat 0.8 sec. a heartbeat. The updated value of CO is fed back to the . The updated value of CO is fed back to the breathing model. breathing model. PASI 2011 - A. Bandoni 73
Computational Implementation Computational Implementation CO 2 , O 2 Cd(inyectable) Cd(inhalable) Cd (alveoli) Transport and Respiratory distribution, system system Pharmacokinetics of Pharmacokinetics of drugs Q a3 , Q p3 Cd (organs) Cardiovascular C di l Ph Pharmacodynamics d i Baroreflex system of drugs MAP E maxBARO E max Eff Emax R a2BARO R sis Eff Ra2 R a3BARO Eff Ra3 Control Action PASI 2011 - A. Bandoni 74
Computational Implementation Computational Implementation Dimensions of the integrated model Var./Eqs. Var./Ecs. Model Parameters Algebraics Algebraics Differenctials Differenctials Cardiovascular-Respiratory 37 39 53 Respiratory-Pharmacodynamic 60 93 85 Total 97 132 138 PASI 2011 - A. Bandoni 75
76 Results Results PASI 2011 - A. Bandoni
Results: cardio vascular system Results: cardio vascular system Wiggers Diagram PASI 2011 - A. Bandoni 77
Results: cardio vascular system Results: cardio vascular system Left ventricular volume Left ventricle and root aortic pressure vs. time vs. time PASI 2011 - A. Bandoni 78
Results: cardio vascular system Results: cardio vascular system Outflow of the left ventricle Outflow of the left ventricle Pressure vs. Volume left ventricle Left ventricular pressure PASI 2011 - A. Bandoni 79
Results: Results: baroreflex baroreflex system system Heart period vs time Heart period vs. time Resist sect A Resist. sect. A s1 of syst. arteries vs. time of syst arteries vs time Compliance in sect. V s1 of sistemic Unstres. Vol. of sect. V s1 of veins vs. time sistemic veins vs. time PASI 2011 - A. Bandoni 80
Results: Results: baroreflex baroreflex system system Comparison of CO vs. time in front of 10 % Sistolic max. elastance of left ventr.vs. time bleeding, with and without baroreceptor Comparison of MAP vs. time in front of 10 % bleeding, with and without baroreceptor PASI 2011 - A. Bandoni 81
Results: gas transport Results: gas transport Partial pressure of O 2 in different compartments of the body Partial pressure of CO 2 in different compartments of the body compartments of the body PASI 2011 - A. Bandoni 82
Results: respiratory system Results: respiratory system expiración expiración inspiración Volume vs. Pressue diagram in lungs Partial pres. profile of O 2 in Partial pres. profile of lung and alveoli. CO 2 in lung and alveoli. PASI 2011 - A. Bandoni 83
Results: Results: pharmacodymic pharmacodymic system system 1µg/kg/min Effect of the SNP action SNP concentration profile at the central arterial compartment Cardiac Output, CO Mean Arterial Pressure, MAP PASI 2011 - A. Bandoni 84
Results: pharmacodynamic system Results: pharmacodynamic system 1µg/kg/min Effect of the SNP action Resistance, R a3 Resistance, R a2 PASI 2011 - A. Bandoni 85
Results: pharmacodynamic system Results: pharmacodynamic system Effect of the DP action 5µg/kg/min DP concentration profile at the central arterial compartment Cardiac Output, CO Mean Arterial Pressure, MAP PASI 2011 - A. Bandoni 86
Results: pharmacodynamic system Results: pharmacodynamic system Effect of the DP action 5µg/kg/min Medial arterial resistances Elastance PASI 2011 - A. Bandoni 87
Results: pharmacodynamic system Results: pharmacodynamic system Effect of the DP action 2, 4, 6, 8 µg/kg/min Volume Index vs. infusion doses (time) Cardiac Index vs. infusion doses (time) Systolic and diastolic pressure vs. infusion doses (time) PASI 2011 - A. Bandoni 88
Results: pharmacodynamic system Results: pharmacodynamic system Effect of the DP action 2, 4, 6, 8 µg/kg/min Cardiac frequency vs. infusion Systemic Resistance vs. infusion doses (time) doses (time) d doses (time) (ti ) PASI 2011 - A. Bandoni 89
Results: pharmacodynamic system Results: pharmacodynamic system Effect of the PFL action Effect of the PFL action 150µg/kg/min / / Mean Arterial Pressure, MAP PFL conc. at the central arterial comp. Cardiac Output, CO Compliance of sector a1 of systemic arteries PASI 2011 - A. Bandoni 90
Conclusions Conclusions Development Development of of an an integrated integrated cardiovascular, cardiovascular, baroreceptor, baroreceptor, respiratory, respiratory, pharmacokinetic pharmacokinetic and and pharmacodynamic pharmacodynamic model model. . The The effect effect of of certain certain drugs drugs on on hemodynamic hemodynamic variables variables was was studied studied. . PASI 2011 - A. Bandoni 91
Future Works Future Works General General model model validation validation with with real real patient patient data data. . Collaboration Collaboration with a with a research research group group formed formed by by doctors doctors ( (Favaloro Favaloro University, University, Bs Bs.As Bs As Bs As As. – Español Español Hospital Español Español Hospital, Hospital B Blanca Hospital, B. Blanca, Blanca, Arg Blanca Arg Arg.) Arg ) Model Model Model validation Model validation validation for validation for for inhalable for inhalable inhalable anesthesia inhalable anesthesia anesthesia effects anesthesia effects effects effects. Model Model Model validation Model validation validation for validation for for simultaneously for simultaneously simultaneously drugs simultaneously drugs drugs administration drugs administration administration. administration Development Development Development of Development of of a control of a control control model control model model for model for for handling for handling handling dose handling dose dose of dose of of drug of drug drug drug administration. administration . Development Development of of a a teaching teaching simulation simulation model model of of the the cardiovascular cardiovascular system system ( (Instituto Instituto Nacional Nacional de de Tecnología Tecnología Industrial, Industrial, INTI, INTI, Bs Bs. .As As. ., , Arg Arg. .) ) PASI 2011 - A. Bandoni 92
Basic References: Cardiovascular Model: Ottesen J., Olufsen M. and Larsen J. Applied Mathematical Models in Ottesen J., Olufsen M. and Larsen J. Applied Mathematical Models in Human Physiology . SIAM, Philadelphia. (2004) Pharmacodynamic Model: Pharmacodynamic Model: Gopinath R., Bequette B., Roy R. and Kaufman H. Issues in the Design of a Multirate Model- based Controller for a Nonlinear Drug Infusion System Biotechnol Prog 11 (3) pp 318 32 (1995) System. Biotechnol. Prog. 11 (3), pp 318–32. (1995) Respiratory Model: Christiansen T. and Dræby C. Modeling the Respiratory System Ch i ti T d D b C M d li th R i t S t Technical. Report IMFUFA, Roskilde University Denmark Text No. 318. (1996) PASI 2011 - A. Bandoni 93
Other References: Dua P and Pistikopulos E Modelling and control of drug delivery systems Comp Dua P and Pistikopulos E. Modelling and control of drug delivery systems . Comp. Chem. Eng. 29 pp. 2290-96. (2005) Montain M Bandoni J y Blanco A Modelado del sistema cardiorespiratorio Montain M, Bandoni J y Blanco A . Modelado del sistema cardiorespiratorio humano: un estudio de simulación. VI CAIQ (Congreso Argentino de Ing. Química) Mar del Plata 26 al 29 de septiembre (2010). Rao R, Bequette B and Roy R. Simultaneous regulation of hemodynamic and anesthetic states: a simulation study ; Annals of Biomedical Engineering, 28 pp. 71- 84. (2000) ( ) Dua P, Dua V and Pistikopoulos E. Modelling and mult-parametric control for delivery of anaesthetic agents. Med. Biol. Eng. Comput. 48 543-53. (2010). Massoud T., G. Eorge, J. Hademenos, W. Young , E. Gao, J. Pile-Spellman and F. Uela. Principles and philosophy of modeling in biomedical research .The FASEB Journal, vol. 12 no. 3, pp.275-285, March 1, 1998. Ottesen J.T. The Mathematical Microscope ‐ Making the inaccessible accessible . Bi Biomedical and Life Sciences Systems Biology ‐ Volume 2, 2011. di l d Lif S i S t Bi l V l 2 2011 PASI 2011 - A. Bandoni 94
“With growing emphasis being placed on the information d processing aspects of biomedical investigation, theoretical and experimental studies assume increasing importance experimental studies assume increasing importance. In many In many instances, however, there are questions that appear to be unanswerable by present experimental techniques; in such cases, y p p q ; , models can usefully augment direct scientific experimentation. The essential ingredient of the scientific method is the use of Th i l i di f h i ifi h d i h f models. Good modeling is more likely to be achieved by following the rules of good thinking However the ideal model cannot be the rules of good thinking. However, the ideal model cannot be achieved. Partial models, imperfect as they may be, are the only means developed by and available to scientists for understanding p y g the universe” Principles and philosophy of modeling in biomedical research. T. Massoud, G. Eorge, J. Hademenos, W. Young , E. Gao, J. Pile-Spellman and F. Uela (University of California at Los Angeles, Columbia University, University of Dallas) The FASEB Journal, vol. 12 no. 3, pp.275-285, March 1, 1998 PASI 2011 - A. Bandoni 95
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97 Muchas gracias PASI 2011 - A. Bandoni
Cámara izquierda del corazón Circulación sistémica dQ p p R Q la la lv la la si p p la lv dQ p p R Q dV a 1 a 1 a 2 a 1 a 1 dt L a 1 la Q Q lv a 1 dt dt L L dt dt a a 1 1 0 Q si p p la la lv V V p p a 1 un , a 1 a 2 a 3 Q p 1 a 2 dV a la R C p E V V a 2 Q Q a 1 la la la d , la l 2 la dt dt V V dV a 2 un , a 2 a 2 p dQ p p Q Q 2 lv lv as a 1 a 2 a si p p C dt lv as a 2 dt L lv 0 p p dV Q si p p as v 1 a 3 lv lv as Q Q Q a 3 2 3 a a R dt a 3 dV lv p E t V V Q Q Q Q lv lv lv lv lv lv d d , , lv lv la la lv lv V V V V p p dt a 3 un , a 3 v 1 v 2 Q p v 1 a 3 R C v 1 a 3 t V Q dt 2 ml lv , b lv * t V V V V dV dV v 1 1 un , v 1 1 v 1 1 p Q Q v 1 a 3 v 1 C dt 1 v E t E 1 t E t lv min, lv max, lv 2 dQ dQ p p p p R R Q Q dV dV v v 2 2 v v 2 2 ra ra v v 2 2 v v 2 2 t t 2 t t v 2 2 Q Q a sin b sin 0 t t v 1 v 2 ce dt L dt v 2 t t t ce ce 0 t t t ce h V V V V v v 2 2 un un , v v 2 2 pv 2 t t p R sQ p ce 0 1 h as 0 lv a 1 C v 2 PASI 2011 - A. Bandoni 98
Circulación pulmonar Cámara derecha del corazón dQ p p R Q dQ p p R Q p 1 p 1 p 2 p 1 p 1 dV ra ra rv ra ra p 1 si p p Q Q ra rv rv p 1 dt L dt L p 1 dt ra 0 V V p p Q si p p p 1 un , p 1 p 2 p 3 ra ra rv p Q 1 p p 2 C R p 1 p 2 dV ra Q Q V V p E V V v 2 ra dV p 2 un , p 2 ra ra ra d , ra p 2 dt p p Q Q Q Q p p 2 2 p p 1 1 p p 2 2 C dt p 2 p p dQ rv ap rv p p si p p ps l 1 dV rv ap p 3 dt L Q Q Q rv p 3 p 2 p 3 R dt p 3 0 Q Q 0 si i p p V V V V rv rv ap p p p 3 un , p 3 l 1 l 2 Q p l 1 p 3 R C l 1 dV p 3 rv p E t V V Q Q rv rv rv d , rv ra rv V V dt dV l 1 un , l 1 l 1 p Q Q l 1 p 3 l 1 t C dt l l 1 1 V Q dt 2 ml rv , b rv * t dQ p p R Q dV l 2 l 2 la l 2 l 2 l 2 Q Q l 1 l 2 dt L dt l 2 E t E 1 t E t min, max, rv rv rv V V V V l l 2 2 un , l l 2 2 pl 2 p R pQ p 0 1 ap rv p C l 2 Modelo respiratorio (fracción molar) Modelo respiratorio (fracción molar) n f R f f f f d I U p I p p T 0 m 0 e 0 i 0 i 0 2 dt R R i 1 0 x 0 0 i C p V p 0 00 0 0 I I x x x x 0 f f f R d T I p U p i 0 t i 0 i κ p f p cp i i 2 dt R i C p V p i 0 i i i PASI 2011 - A. Bandoni 99
Barorreceptores 1 1 n MAP n MAP p s 1 c c c MAP Q Q p MAP d d c c 1 1 1 1 2 2 c c c c p p b b 1 1 V V Q Q x b b p x Q Q dt d 1 2 b MAP n MAP n MAP i s i p i i i E H , E , R , V , C max ps un v 0 0 dx t 1 b i x t MAP , i E i i M dt i CO c 2 M O M M 0 2 Modelo respiratorio (presión) O 2 c O O 0 2 2 c c aa M aa n c dpi aa aa U p R C m 0 0 i p p U dpi dp dt 0 i t 0 i 1 dt R C dt R C 0 0 i i 3 3 2 2 H a H a 1 H a 0 0 2 0 . 27273 1 . 96364 t 0 x 0 . 278 a K NaOH K 0 . 66943 0 . 53554 t 0 . 278 x 1 . 806 0 2 a , Pr a , CO U m 17 . 00005 8 . 50909 t 1 . 806 x 1 . 904 a K K NaOH N OH c K K K K N OH NaOH H H Pr P 1 a , CO 0 CO a , Pr a , CO 0 0 0 . 57034 0 . 05904 t 1 . 904 x 5 2 a K K NaOH H Pr c Modelo de transporte de gases en sangre 0 a , CO a , Pr 0 0 CO 2 c c c c 1 p c c d d d d d d c b p c Ery p c Pla p Hb Hb , pH , pH , pH 1 c c p M p M p t b V V z Q CO CO CO as b Ery Ery t p b p i 2 2 c 2 c dt d d Hb Hb Ery p Ery p c Ery p Ery p pH , pH pK , pH , pH 1 10 1 CO CO c p p c 2 2 CO 2 d d c c p p p b b V V Q Q 1 1 s b A b v p dt d Ery p Ery p c p p Ery Ery pH , pH pK , pH , 1 10 pH CO CO c 2 2 CO 2 PASI 2011 - A. Bandoni 100
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